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Abstract:

A method to obtain gain-corrected measurements. A measurement tool having
one or more arrays is provided, wherein the arrays include two co-located
triaxial transmitters and two co-located triaxial receivers. Measurements
are obtained using the transmitters and the receivers. Impedance matrices
are formed from the obtained measurements and the impedance matrices are
combined to provide gain-corrected measurements. The apparatus may
alternatively be a while-drilling logging tool having one or more arrays,
wherein each array comprises a transmitter, a receiver, and a buck, and
wherein the signal received by the receiver is subtracted from the signal
received by the buck or vice versa. A slotted shield may be incorporated
into either embodiment of the tool. The slots may form one or more island
elements. A material is disposed in the slots. The islands and shield
body have complementary tapered sides that confine the islands within the
shield body.

Claims:

1. A method to obtain gain-corrected measurements, comprising: providing
a measurement tool having one or more arrays, wherein the arrays comprise
two co-located triaxial transmitters and two co-located triaxial
receivers; obtaining measurements using the two co-located triaxial
transmitters and the two co-located triaxial receivers; forming impedance
matrices from the obtained measurements; and combining the impedance
matrices to provide gain-corrected measurements.

2. The method of claim 1, wherein the measurement tool comprises a
downhole logging tool, and the providing further comprises disposing the
downhole logging tool on a wireline, a drill string, or a wired drill
pipe.

3. The method of claim 1, wherein the receivers are located between the
transmitters and the distance from one of the transmitters to one of the
receivers is substantially equal to the distance from the other
transmitter to the other receiver.

4. The method of claim 1, wherein the transmitters are located between
the receivers and the distance from one of the transmitters to one of the
receivers is substantially equal to the distance from the other
transmitter to the other receiver.

5. The method of claim 1, wherein the obtaining measurements further
comprises successively broadcasting a signal from each of the antennas
comprising the transmitters.

6. The method of claim 1, wherein the obtaining measurements comprises
transmitting and receiving a plurality of electromagnetic signals, each
of the electromagnetic signals being transmitted at different
frequencies.

8. The method of claim 1, wherein the measurements are obtained while the
measurement tool is rotating or sliding.

9. The method of claim 1, wherein one of the one or more triaxial
transmitters has antennas that are azimuthally offset from antennas
comprising one or more of the other triaxial transmitters and/or
receivers.

10. The method of claim 1, wherein one or more of the two or more
triaxial transmitters and/or one or more of the two or more triaxial
receivers each have antennas that are azimuthally separated by 120
degrees and tilted relative to a longitudinal axis of the measurement
tool.

11. The method of claim 10, wherein the tilted antennas are tilted at an
angle of approximately arctan 2.

12. The method of claim 1, further comprising constructing calibrated
measurements using the gain-corrected measurements.

13. The method of claim 1, wherein the forming impedance matrices further
comprises arranging the measurements in three-by-three matrices.

14. The method of claim 13, wherein thirty-six measurements are arranged
into four matrices.

16. An apparatus to determine a formation property of a subsurface
formation, comprising: a while-drilling logging tool having one or more
arrays, wherein each array comprises a transmitter, a receiver, and a
buck, and wherein the signal received by the receiver is subtracted from
the signal received by the buck or vice versa.

17. A shield for a downhole logging tool, comprising: a shield body
having slots therein; one or more island elements bordered by the slots;
and a material disposed in the slots; wherein the islands and shield body
have complementary tapered sides that confine the islands within the
shield body.

18. The shield of claim 17, wherein the shield body is non-conductive but
the material disposed in the slots in conductive.

19. The shield of claim 17, wherein the shield body is conductive but the
material disposed in the slots in non-conductive.

20. The shield of claim 17, wherein the shield is substantially
electromagnetically transparent.

21. The shield of claim 17, wherein the downhole logging tool is an
induction or propagation-type while-drilling logging tool.

22. A method to obtain an impedance matrix using a substantially
non-rotating downhole tool, comprising: providing the downhole tool which
has one co-located triaxial transmitter and one co-located triaxial
receiver; obtaining measurements using the co-located triaxial
transmitter and the co-located triaxial receiver; determining the
components of the impedance matrix from the obtained measurements.

23. The method of claim 22, wherein components of the impedance matrix
are combined to provide gain-corrected measurements.

24. The method of claim 22, wherein measured gains are used to provide
gain-corrected measurements.

25. The method of claim 22, wherein the triaxial transmitter and/or the
triaxial receiver have antennas that are azimuthally separated by 120
degrees and tilted relative to a longitudinal axis of the downhole tool.

26. The method of claim 25, wherein the tilted antennas are tilted at an
angle of arctan 2.

[0003] The present disclosure relates generally to the logging of
subsurface formations surrounding a wellbore using a downhole logging
tool, and particularly to obtaining gain-corrected measurements.

[0004] 2. Background Art

[0005] Logging tools have long been used in wellbores to make, for
example, formation evaluation measurements to infer properties of the
formations surrounding the borehole and the fluids in the formations.
Common logging tools include electromagnetic tools, nuclear tools, and
nuclear magnetic resonance (NMR) tools, though various other tool types
are also used.

[0006] Early logging tools were run into a wellbore on a wireline cable,
after the wellbore had been drilled. Modern versions of such wireline
tools are still used extensively. However, the need for information while
drilling the borehole gave rise to measurement-while-drilling (MWD) tools
and logging-while-drilling (LWD) tools. MWD tools typically provide
drilling parameter information such as weight on the bit, torque,
temperature, pressure, direction, and inclination. LWD tools typically
provide formation evaluation measurements such as resistivity, porosity,
and NMR distributions. MWD and LWD tools often have components common to
wireline tools (e.g., transmitting and receiving antennas), but MWD and
LWD tools must be constructed to not only endure but to operate in the
harsh environment of drilling.

SUMMARY

[0007] A method to obtain gain-corrected measurements. A measurement tool
having one or more arrays is provided, wherein the arrays include two
co-located triaxial transmitters and two co-located triaxial receivers.
Measurements are obtained using the transmitters and the receivers.
Impedance matrices are formed from the obtained measurements and the
impedance matrices are combined to provide gain-corrected measurements.
The apparatus may alternatively be a while-drilling logging tool having
one or more arrays, wherein each array comprises a transmitter, a
receiver, and a buck, and wherein the signal received by the receiver is
subtracted from the signal received by the buck or vice versa. A slotted
shield may be incorporated into either embodiment of the tool. The slots
may form one or more island elements. A material is disposed in the
slots. The islands and shield body have complementary tapered sides that
confine the islands within the shield body.

[0008] Other aspects and advantages will become apparent from the
following description and the attached claims.

[0016] FIG. 8 schematically shows triaxial magnetic moments m1, m2, and m3
about a collar, where the coils are oriented 120 degrees apart
azimuthally and dipping at an angle 54.74 degrees off the tool axis, in
accordance with the present disclosure.

[0017]FIG. 9 is a diagram of an exemplary slot pattern of an unwrapped
shield for a set of co-located triaxial coils, in accordance with the
present disclosure.

[0048]FIG. 40 shows an exemplary tilted test coil on an LWD tensor
resistivity tool with antenna spacings, in accordance with the present
disclosure.

DETAILED DESCRIPTION

[0049] Some embodiments will now be described with reference to the
figures. Like elements in the various figures will be referenced with
like numbers for consistency. In the following description, numerous
details are set forth to provide an understanding of various embodiments
and/or features. However, it will be understood by those skilled in the
art that some embodiments may be practiced without many of these details
and that numerous variations or modifications from the described
embodiments are possible. As used here, the terms "above" and "below",
"up" and "down", "upper" and "lower", "upwardly" and "downwardly", and
other like terms indicating relative positions above or below a given
point or element are used in this description to more clearly describe
certain embodiments. However, when applied to equipment and methods for
use in wells that are deviated or horizontal, such terms may refer to a
left to right, right to left, or diagonal relationship as appropriate.

[0050] FIG. 1 illustrates a well site system in which various embodiments
can be employed. The well site can be onshore or offshore. In this
exemplary system, a borehole 11 is formed in subsurface formations by
rotary drilling in a manner that is well known. Some embodiments can also
use directional drilling, as will be described hereinafter.

[0051] A drill string 12 is suspended within the borehole 11 and has a
bottom hole assembly 100 which includes a drill bit 105 at its lower end.
The surface system includes platform and derrick assembly 10 positioned
over the borehole 11, the assembly 10 including a rotary table 16, kelly
17, hook 18 and rotary swivel 19. The drill string 12 is rotated by the
rotary table 16, energized by means not shown, which engages the kelly 17
at the upper end of the drill string. The drill string 12 is suspended
from a hook 18, attached to a traveling block (also not shown), through
the kelly 17 and a rotary swivel 19 which permits rotation of the drill
string relative to the hook. As is well known, a top drive system could
alternatively be used.

[0052] In the example of this embodiment, the surface system further
includes drilling fluid or mud 26 stored in a pit 27 formed at the well
site. A pump 29 delivers the drilling fluid 26 to the interior of the
drill string 12 via a port in the swivel 19, causing the drilling fluid
to flow downwardly through the drill string 12 as indicated by the
directional arrow 8. The drilling fluid exits the drill string 12 via
ports in the drill bit 105, and then circulates upwardly through the
annulus region between the outside of the drill string and the wall of
the borehole, as indicated by the directional arrows 9. In this well
known manner, the drilling fluid lubricates the drill bit 105 and carries
formation cuttings up to the surface as it is returned to the pit 27 for
recirculation.

[0053] The bottom hole assembly 100 of the illustrated embodiment includes
a logging-while-drilling (LWD) module 120, a measuring-while-drilling
(MWD) module 130, a roto-steerable system and motor, and drill bit 105.

[0054] The LWD module 120 is housed in a special type of drill collar, as
is known in the art, and can contain one or a plurality of known types of
logging tools. It will also be understood that more than one LWD and/or
MWD module can be employed, e.g. as represented at 120A. (References,
throughout, to a module at the position of 120 can alternatively mean a
module at the position of 120A as well.) The LWD module includes
capabilities for measuring, processing, and storing information, as well
as for communicating with the surface equipment. In the present
embodiment, the LWD module includes a resistivity measuring device.

[0055] The MWD module 130 is also housed in a special type of drill
collar, as is known in the art, and can contain one or more devices for
measuring characteristics of the drill string and drill bit. The MWD tool
further includes an apparatus (not shown) for generating electrical power
to the downhole system. This may typically include a mud turbine
generator powered by the flow of the drilling fluid, it being understood
that other power and/or battery systems may be employed. In the present
embodiment, the MWD module includes one or more of the following types of
measuring devices: a weight-on-bit measuring device, a torque measuring
device, a vibration measuring device, a shock measuring device, a
stick/slip measuring device, a direction measuring device, and an
inclination measuring device.

[0056] An example of a tool which can be the LWD tool 120, or can be a
part of an LWD tool suite 120A, is shown in FIG. 2. As seen in FIG. 2,
upper and lower transmitting antennas, T1 and T2, have upper
and lower receiving antennas, R1 and R2, therebetween. The
antennas are formed in recesses in a modified drill collar and mounted in
insulating material. The phase shift of electromagnetic energy as between
the receivers provides an indication of formation resistivity at a
relatively shallow depth of investigation, and the attenuation of
electromagnetic energy as between the receivers provides an indication of
formation resistivity at a relatively deep depth of investigation. U.S.
Pat. No. 4,899,112 can be referred to for further details. In operation,
attenuation-representative signals and phase-representative signals are
coupled to a processor, an output of which is coupleable to a telemetry
circuit.

[0057] Recent electromagnetic logging tools use one or more tilted or
transverse antennas, with or without axial antennas. Those antennas may
be transmitters or receivers. A tilted antenna is one whose dipole moment
is neither parallel nor perpendicular to the longitudinal axis of the
tool. A transverse antenna is one whose dipole moment is substantially
perpendicular to the longitudinal axis of the tool, and an axial antenna
is one whose dipole moment is substantially parallel to the longitudinal
axis of the tool. A triaxial antenna is one in which three antennas
(i.e., antenna coils) are arranged to be mutually independent. That is,
the dipole moment of any one of the antennas does not lie in the plane
formed by the dipole moments of the other two antennas. Three orthogonal
antennas, with one antenna axial and the other two transverse, is one
example of a triaxial antenna. Two antennas are said to have equal angles
if their dipole moment vectors intersect the tool's longitudinal axis at
the same angle. For example, two tilted antennas have the same tilt angle
if their dipole moment vectors, having their tails conceptually fixed to
a point on the tool's longitudinal axis, lie on the surface of a right
circular cone centered on the tool's longitudinal axis and having its
vertex at that reference point. Transverse antennas obviously have equal
angles of 90 degrees, and that is true regardless of their azimuthal
orientations relative to the tool.

[0058] One possible embodiment of antenna design includes multi-component
coils. For example, a co-located triaxial tilted antenna used for
downhole resistivity measurements may be provided. The tilted coils each
comprise a portion of a closed circuit around the collar perimeter, and
can be either embedded in a recess about the tool collar or in a
nonconductive cylinder that slides over the collar. The design has at
least one triaxial antenna that can be used as a transmitter (or a
receiver) with at least one additional antenna displaced along the tool
axis as a receiver (or a transmitter). Multiple antennas with different
spacings and frequencies may be used to cover the desired conductivity
ranges and depths of investigation. The effect of farther
transmitter-receiver spacing and/or a more conductive formation is
compensated by using a lower frequency signal. FIG. 3 shows a tool with a
tilted transmitter and a triaxial orthogonal receiver (the three
receivers could also be orthonormal to one another). FIG. 4 shows the
magnetic dipole equivalent of a logging tool with a co-located triaxial
tilted transmitter and a tilted receiver, and FIG. 5 shows another
embodiment in which the transmitter and the receiver are both co-located
triaxial tilted antennas.

[0059] It can be shown that the square of the norm of raw measurements
between a transmitter, T, and a receiver, R, is a function of the
instantaneous tool face angle, and that it can be decomposed into a
finite set of Fourier coefficients. Below is the general formula of the
coupling between two magnetic dipoles with known orientations, as
depicted in FIG. 6. FIG. 6 shows a T and an R antenna both tilted so that
their equivalent magnetic dipole moments are at an angle β relative
to the tool axis (45 degrees for FIG. 6). However, as shown, the antennas
may have different azimuthal orientations. The azimuthal orientation of T
relative to R is denoted by an angle α. That is, α is the
angle between the projection of the receiver dipole moment onto the
tool-fixed xy-plane and the projection of the transmitter dipole moment
onto the same (or a parallel) plane. During drilling, if the tool
rotates, for example, by a tool face angle φ, the T and R magnetic
moments will rotate along with the tool while measurements are performed.

where the components of the coupling matrix (in the middle) (ij) are the
elementary measurements in the absence of rotation when a transmitter in
the i direction and a receiver in the j direction are used. The two
matrices multiplying the coupling matrix are the rotation matrices that
account for tool face angle. Finally, the vector on the right hand side
is the orientation of the R dipole moment, while the one on the left hand
side is that of the T antenna. The three matrixes in the middle can be
re-written as M leading to:

[0061] From the preceding equations, it is apparent that voltages measured
at receivers will be periodic functions of the tool face angle φ and
2φ. The measured voltage can be represented as the second order
Fourier expansion given by:

VTR(φ)=a+bcos(φ)+csin(φ)+dcos(2φ)+esin(2φ)

Those coefficients are simple linear combinations of individual terms of
the coupling tensor. The relationships between the Fourier coefficients
and the tensor coefficients are given by the next set of equations (after
normalization by a factor of two):

The extraction of Fourier coefficients is actually a linear problem,
since the measured voltage at the receiver is a linear function of the
unknown vector x=[a,b,c,d,e] with a known vector w expressed as:

w=[1, cos(φ), sin(φ), cos(2φ), sin(2φ)]T

[0062] Once at least five measurements are performed at different angles,
the vector of Fourier coefficients can be computed by a Least-Square fit:

[0063] The symbol ξ corresponds to a good estimate of noise standard
deviation. To estimate ξ, the standard deviation of noise, we compute
residuals on a sliding window of past points versus prediction based on
computed Fourier coefficients by applying the following algorithm:

This estimate is robust through the usage of median filtering on a set of
past observed values, and is made adaptive by the exponential weight used
in the update formula.

[0064] Once the Fourier coefficients are estimated, calibrated
measurements can be constructed. The following shows how different kinds
of measurements can be computed. The descriptions are split based on
harmonics, because each harmonic leads to a measurement having a
different azimuthal sensitivity. DC terms lead to measurements that do
not depend on azimuth, first harmonic terms lead to measurements having a
cos(φ) sensitivity, and second harmonic terms lead to measurements
having a cos(2φ) sensitivity.

[0068] A triaxial co-located orthonormal antenna, where the magnetic
moments shown in FIG. 8 are oriented about the axis of the metal collar,
is suitable for LWD use. These moments can be skewed or non-orthogonal.
The most convenient construction is where the magnetic moments m1, m2,
and m3 are orthonormal, separated by 120 degrees about the z-axis, and
tilted at an angle of arctan(sqrt(2)) (54.74 degrees). The coils are
assumed to be imbedded in a non-conductor within a recess in the collar.
This assembly of orthonormal co-located coils is then protected by a
slotted metal shield. The coils can be recessed in the collar with a
shield fixed over them, or the coils can be embedded in a non-conductive
tube that is inserted into the shield itself. A method of designing such
shield and antenna configurations is described below.

[0069] For the purpose of these discussions, we will concentrate on the
magnetic dipole equivalent of a tilted coil. For convenience, it we
assume the windings of a tilted coil are in a plane, it can be
characterized by two angles. In the description here, the tilt angle is
defined as the angle between the normal to the plane and a transverse
axis (x or y for example). As such the tilt angle relative to the z axis
is 90-β, as shown in FIG. 7. Note that the normal to the antenna
plane is the equivalent magnetic dipole of the antenna. The second angle
is the standard azimuthal angle, φ, used in the polar coordinate
system and is the angle between the x axis and the projection of the
normal onto the xy-plane. With these definitions, the trajectory or
equations of a tilted coil winding are:

x=Rcos φ

y=Rsin φ

z=Rtan βcos φ,

where 90-β is the tilt angle of the coil with respect to the z axis,
R is the radius of the coil, and φ is the azimuthal angle.

[0070] The shield is a cylindrical structure that encompasses the tilted
coil. It contains a series of cut outs or slots to allow electromagnetic
radiation to pass through the metallic shield, as shown in FIG. 9. The
location of the slots may be equally spaced along the trajectory of the
coil. That would functionally make the arc length between any two slots
equal to:

Here the projection of the slot height or length along the tool axis is
set equal to hs.

[0073] As shown in the embodiment of FIG. 10, biaxial co-located antennas
tilted 45 degrees with respect to the tool axis may be wrapped on a
recessed metal collar. The two coils are azimuthally offset by 180
degrees from each other, but the azimuthal offset is not limited to 180
degrees. FIG. 11 shows triaxial co-located tilted antennas that are
azimuthally offset by 120 degree from each other. The supporting metal
collar is preferably recessed as shown in FIG. 11. To have a signal loss
less than 2 dB, the recess width is preferably about 8 times greater than
the recess height.

[0074]FIG. 11 also shows a calibration coil. The calibration coil
provides a simple way to calibrate the antennas simultaneously. A small
current sent to the calibration coil generates a magnetic field. The
co-located antennas receive this magnetic field and generate induced
currents that are proportional to their efficiencies. Thus, the induced
currents provide calibration factors for the tilted coils.

[0075] As mentioned above, shields are cylindrical structures with slots.
If the structure is conductive (metallic), then the slots are
non-conductive and vice versa. For a metallic shield enclosing a coil,
the slots are distributed around the circumference of the shield and they
are cut to be perpendicular to the coil wire. The number of slots is a
design variable. Increasing the number of slots reduces the attenuation
of the radiation through the slots, but as the number of slots increases,
the mechanical integrity of the shield is reduced and, above four or five
slots, the gain in attenuation is not as great.

[0076] FIGS. 12a-12c show the effect of varying the number of slots for
three co-located coils that are tilted at 54.74 degrees relative to the
tool axis (which gives a set of three orthogonally aligned antennas) and
are distributed 120 degrees azimuthally. In FIGS. 12a-12c the three coils
are shown as sinusoidal curves with their corresponding slots. With six
slots (FIG. 12A), only two slots intersect and the shield has good
mechanical integrity. When the number of slots is increased to ten, as in
FIG. 12B, up to four slots can intersect and create a diamond shape that
is not physically connected to the remainder of the metal structure. In
FIG. 12B, there are six diamond shaped cut outs that we call "islands".
These cut out islands need to be kept in the shield structure for both
electrical and mechanical reasons. A method of achieving this is to taper
the edges of the island piece and the associated shield so that the
island's outer surface dimensions are smaller than the shield opening
while the island's inner surface dimensions are larger than the shield
opening. FIG. 13 shows a cross sectional view of this arrangement. Since
the slots are filled with non-conductive material such as epoxy, the
pieces are held together. FIG. 12c shows twelve slots which not only
create islands, but also a complete cut around the circumference of the
shield, which can be detrimental to the mechanical structure of the
shield.

[0077] Another design parameter is the length of the slots. Increasing the
length of the slots improves the efficiency of the antenna. However,
above certain slot lengths the improvement is marginal at best. As with
the higher number of slots, the longer slot length reduces the mechanical
integrity of the shield and can lead to islands. FIGS. 14a-14c show the
effect of varying the slot length for three co-located antennas with
54.74 degree tilt relative to the tool axis and 120 degree azimuthal
offsets. As FIG. 14A shows, with slot lengths of 3 inches, some of the
slots intersect, but only in pairs, so that there are no islands. When
the slot length is increased to four inches, as in FIG. 14B, the
structure comes very close to forming islands without actually doing so.
However, the connections may not be strong enough for mechanical reasons
and provisions such as that shown in FIG. 13 may be used to enhance the
mechanical integrity of the shield. As the slot length increases to 6
inches, as in FIG. 14c, formation of islands is unavoidable.

[0078]FIG. 15 is another example of a shield slot pattern. In this
embodiment, each coil is tilted 45 degrees with respect to the tool axis.
The three co-located antennas are azimuthally rotated by 120 degrees
relative to each other although the offset angle is not limited to 120
degrees. Note that the vertical extent of this shield is less than that
of FIG. 12A or 14a. This is due to the tilt angle.

[0079] It was noted above that a preferred antenna configuration is one in
which three co-located antenna coils are azimuthally rotated by 120
degrees and tilted at an angle of arctan 2 (which is approximately 54.74
degrees). In that case, the vector potential of the magnetic field of a
tilted coil at a point sufficiently far away, i.e., at a distance r, from
the magnetic source can be expanded into an infinite series involving
inverse powers of that distance r. Higher power terms are generally
neglected. If the first three terms of the expansion are kept, the third
term is found to be zero at the particular angle arctan 2. Thus, dipole
coils tilted at that angle can produce a cleaner dipole field.

[0080] As alluded to above, alternative embodiments for measuring the LWD
triaxial resistivity tool response are possible. Certain tool
configurations allow for the generation of one or more combinations of
tool responses that remove the gains of the receivers and the
transmitters. One such tool, a triaxial propagation tool, preferably
operates at multiple frequencies, in the MHz range, to cover the
conductivity range from 0.1 ohm-m to 1000 ohm-m. However, such a tool
potentially has a limited depth of investigation and limited conductivity
range per frequency. This is not an ideal configuration for geo-steering,
but may be adequate for formation evaluation near the tool. This response
may be inverted for Rh, Rv, dip, azimuth, and bed thickness. This
information may be used to build a formation model for inputs to the
lower frequency, longer spacing tool described above. Each measurement
spacing will involve two receiver antennas and two transmitters.

[0081] An alternative tool configuration comprises a triaxial induction
tool. The triaxial induction tool generally comprises multiple main
(transmitters and receivers) coils and bucking coils, all spaced along
the tool's longitudinal axis, and generally operates at a single
frequency, typically around 25 kHz. The induction tool typically has one
or more arrays, wherein each array comprises a transmitter, a receiver,
and a buck, and wherein the signal received by the receiver is subtracted
from the signal received by the buck or vice versa. In an LWD environment
borehole corrections and invasion information are not needed to correct
the raw measurements of the deeper measurements, thus one would need less
spacing. The resistivity range of operation for an induction measurement
is generally from 0.1 to 500 ohm-m, and the depth of investigation will
be on the order of the spacing. This measurement is ideal for
geo-steering, formation geology, and formation evaluation. Each
measurement spacing will involve two receiver antennas and a single
transmitter antenna. This yields a net decrease in the number of antennas
compared to the propagation measurements since only one triaxial
transmitter will be needed for all the triaxial receiver spacings.

[0082] Various techniques exist for making measurements using magnetic
dipole moment transmitters and receivers for a transverse anisotropic
medium with plane-parallel layers that are transversely isotropic (TI
anisotropy). Preferably, to make such measurements, the thickness of the
bed is greater than the transmitter-receiver spacing for a given
transmitter-receiver pair. For example, for a transmitter carrying a
current I, the voltage V measured at the receiver can be expressed in
terms of tensor-transfer impedance ZRT:

V=IuR ZRTuT, Eq 2

where uR and uT are a unit vectors along the receiver and
transmitter coil axes, respectively. The transfer impedance ZRT has
the following symmetry property:

ZRT= ZTRT,

where the superscript T denotes the transpose.

[0083] Two sets of orthogonal unit vectors are introduced ux,
uy, uz, for the formation, and uX, uY, uZ, for
the tool coordinates, with uZ along the axis of symmetry of the
tool. The z-axis is perpendicular to the layers, oriented upward. The
tool axis is confined to the x-z plane (i.e., the formation azimuth is
zero). The formation dip angle is denoted by α, so that the
formation system with respect to the tool system is given by:

uX=ux cos α+uz sin α

uY=uy

uZ=-ux sin α+uz cos α Eq. 3

[0084] The symmetrized measurement in the tool coordinates can be
transformed or rotated to formation coordinates as follows:

Note that all the off-diagonal terms with the subscript Y are zero
due to the tool being confined to the xz-plane. Now we can express the
voltage in the formation coordinates for all nine terms of the tensor:

[0085] For the triaxial co-located tool configuration shown in FIG. 16,
which has two transmitters and two receivers placed symmetrical about the
tool origin and along its axis, we can express the transfer impedance for
the uphole transmitter T1 and uphole receiver R1 as:

z11= GT1 Z11 GR1, Eq. 7

where GT1 and GR1 are the diagonal complex gain matrices for
T1 and R1, respectively, and Z11 is the transfer impedance
for T1 and R1, respectively. Similarly, we can express the
transfer impedance for other possible combinations:

z12= GT1 Z12 GR2,

z21= GT2 Z21 GR1, and

z22= GT2 Z22 GR2.

Next, we can combine these measurements as the product of the near,
inverse transfer impedance and the far transfer impedance for a
downwardly propagating wave:

Td=( z11)-1 z12

and for an upwardly propagating wave:

Tu=( z12)-1 z21.

To remove the sensor gains resulting from sensor geometry and electronic
variation, we can combine Td and the transpose of Tu term by
term:

M1(α)=Td*TuT=( z11)-1 z12*[(
z22)-1 z21]T Eq. 8

For the special case of Eq. 8 when the relative dip is zero or where a=0
in Eq 6, we have:

Note that the ZZ-term is just the usual axial response upon taking the
logarithm Likewise, the other diagonal terms could be handled in the same
fashion to remove the undesired gains. For the special case of Eq. 8 when
the relative dip is non-zero in Eq. 6, we have:

[0086] In this case, the XX and ZZ terms are more complex, but the
attenuation and phase responses of these terms are as expected. The XZ
and ZX terms do not behave quite as expected since we multiplied by the
transpose of the Tu term, however those terms do have large
responses when approaching a horizontal bed at high dip.

[0087] There are many ways to manipulate these tensors and another option
is to matrix multiply Td and Tu:

For the special case of Eq. 8 when the relative dip is non-zero in Eq. 6,
we have:

[0088] There we see that the off-diagonal terms have a receiver gain ratio
that can be measured using the fact that the tool rotates. Thus, every 90
degree rotation of the tool, the gain ratio of gx/gz is equal to gy/gz
and so on. Alternatively, we can multiply the xz-term by the zx-term.

[0089] We can calculate the attenuation and phase from the formulation:

This is obtained by taking the matrix natural log of the square root of
Eq. 9 To do that, we first perform the element multiplication in Eq. 8.
Then we take the element square root, and finally the matrix natural
logarithm to determine a harmonic average for borehole compensation.

[0090] We modeled the triaxial tool shown in FIG. 16. Plots of the
elements of Eq. 10 and the tensor with azimuth set to zero with varying
Rh, Rv, and dip were studied using a point dipole formulation for
transfer impedance for a formation with dip, azimuth, Rh, and Rv. See
FIGS. 17-24 for some characteristic responses of . The attenuation and
phase of tensor was modeled for the case of azimuth=0 degrees. The tool
was modeled for frequencies of 400 kHz and 2 MHz with transmitter-center
receiver spacing of 30 (36) inches and a receiver to receiver spacing of
12 (6) inches. The attenuation and phase have good sensitivity versus
anisotropy and dip, but these are shallow measurements due to the high
frequencies in skin effect contribution to the voltage.

[0091] The tool attenuation and phase shift response using while logging
through a three-bed formation having varying dip and anisotropy can be
modeled. A simple exemplary formation model is shown in FIG. 25. The
resistivity attenuation and phase transformations are plotted for the
tool operating at 400 kHz and 2 MHz in an infinite homogeneous formation
in FIGS. 26-29. Note that the XX resistivity transforms are doubled
valued, therefore we will only make the transforms from the low
resistivity to the minimums.

[0092] Next we plot the responses of the tool logging through three beds
at a dip of 60 degrees and as a function of anisotropy ratios of 1, 2, 5,
10, and 20. The diagonal terms XX, YY, and ZZ are in units of resistivity
or ohm-m, while the XZ and ZX terms are in units of dB. The resistivity
responses are shown in FIGS. 30-33.

[0093] Next we plot the responses of the tool logging through three beds
as a function of dip for an anisotropy ratio of two. The diagonal terms
XX, YY, and ZZ are in units of resistivity or ohm-m, while the XZ and ZX
terms are in units of dB. The resistivity responses are shown in FIGS.
34-37.

[0094] The tensor responses for the triaxial induction tool in wireline
are well known. We can also measure the apparent conductivity tensor
σappk for the k-th spacing and invert a 1D-dipping
layered earth model for the Rh_k, Rv_k, dip_k, azi_k, and bed thickness:

[0095] The calibration of the triaxial or tensor resistivity tool on the
LWD platform for i-th transmitter and the j-th receiver and the k-th
spacing cab be functionally expressed as:

σijkapp=gelec(Te)gijkTTL(σ.sub-
.ijkmeas-σijkSEC(Ta))

where σijkapp is the calibrated complex apparent
conductivity and gijkTTL is the gain correction defined for a
modeled reference tilted test loop as:

g ij TTL = σ ijk TTL Ref σ ijk TTL Meas
. ##EQU00028##

The modeled tilted test loop response is given by
σijkTTL.sup.Ref and the measured tilted test loop
response is given by σijkTTL.sup.Meas. The tilted test
loop is shown in FIG. 40 as it is logged over a triaxial induction LWD
tool or put at specified axial and azimuthal positions. The phase
correction is:

where ηjk is the electronics gain/phase correction, Kijk is
a sensitivity factor, VRjk is the voltage on the receiver, and
ITi is the transmitter current. The background correction,
σijkSEC(T), is given by:

σ ijk SEC ( T ) = η jk K ijk V jk SEC
( T ) I i ( T ) . ##EQU00031##

A test loop is used to either transmit or receive a test signal for each
transmitter, receiver, and bucking coil on an LWD induction tool. The
gain can then be determined for each of those antennas. The temperature
offset is acquired by slowly heating the tool and then fitting the tool
response to a nth-order polynomial fit. The coefficients are then stored
downhole, as are the gains, to correct or calibrate the tool's raw
measurements. Thus, the gain-corrected receiver signal and the
gain-corrected buck signal can be subtracted one from the other to
provide an LWD induction measurement. The suggested LWD tenser
resistivity tool with three spacings is shown below in FIG. 39. A typical
tool response to a zero azimuth formation versus dip and anisotropy is
shown in FIG. 38. Again, there is good sensitivity to anisotropy and dip.

[0097] It should be appreciated that while the invention has been
described with respect to a limited number of embodiments, those skilled
in the art, having benefit of this disclosure, will appreciate that other
embodiments can be devised which do not depart from the scope of the
invention as disclosed herein. Accordingly, the scope of the invention
should be limited only by the attached claims.

Patent applications by Dean M. Homan, Sugar Land, TX US

Patent applications by Emmanuel Legendre, Sevres FR

Patent applications by Gerald N. Minerbo, Missouri City, TX US

Patent applications by Jean Seydoux, Houston, TX US

Patent applications by Reza Taherian, Sugar Land, TX US

Patent applications by Robert C. Smith, Houston, TX US

Patent applications by SCHLUMBERGER TECHNOLOGY CORPORATION

Patent applications in class By induction or resistivity logging tool

Patent applications in all subclasses By induction or resistivity logging tool